Pseudopotential formalism for fractional Chern insulators

Recently, generalizations of fractional quantum Hall (FQH) states known as fractional quantum anomalous Hall or, equivalently, fractional Chern insulators states, have been realized in lattice models. Ideal wave functions such as the Laughlin wave function, as well as their corresponding trial Hamiltonians, have been vital to characterizing FQH phases. The Wannier function representation of fractional Chern insulators recently proposed [X.-L. Qi, Phys. Rev. Lett. 107, 126803 (2011)] defines an approach to generalize these concepts to fractional Chern insulators. In this paper, we apply the Wannier function representation to develop a systematic pseudopotential formalism for fractional Chern insulators. The family of pseudopotential Hamiltonians is defined as the set of projectors onto asymptotic relative angular momentum components which forms an orthogonal basis of two-body Hamiltonians with magnetic translation symmetry. This approach serves both as an expansion tool for interactions and as a definition of positive-semidefinite Hamiltonians for which the ideal fractional Chern insulator wave functions are exact null-space modes. We compare the short-range two-body pseudopotential expansion of various fractional Chern insulator models at filling $\mu =\frac{1}{3}$ in phase regimes where a Laughlin-type ground state is expected to be realized. We also discuss the effect of inhomogeneous Berry curvature which leads to components of the Hamiltonian that can not be expanded into pseudopotentials, and elaborate on their role in determining low-energy theories for fractional Chern insulators. Finally, we generalize our Chern pseudopotential approach to interactions involving more than two bodies with the goal of facilitating the identification of non-Abelian fractional Chern insulators.

Figure 1

Shift of the Wannier polarization as $k_y$ varies. The azimuthal direction represents the real space $x$, while the poloidal direction represents the periodic domain of $k_y$. For any full rotation along $y$, i.e., $k_y\to k_y+2\pi$, the Wannier state shifts by $x\to x+1$.

Figure 2

Wannier center evolution of the Dirac model used in Ref. 25 from $x=0$ to 10 as a function of (a) $k_y/2\pi$ and (b) of the extended wave vector $K/2\pi$. From (a) to (b), it becomes visible how the evolution of $x$ changes as a function of $K$, suggesting its similarity to Landau level wave functions (c).

Figure 3

Lattice structure of the CB (left) and HC (right) models. Sites colored differently belong to different sublattices. The unit cells are demarcated in red. For the CB model, NN interactions are between different sublattices while NNN and NNNN interactions occur within the same sublattice. For the HC model, both the NN and NNNN interactions occur between different sublattices, but the NNN interactions act within the same sublattice.

Figure 4

The pseudopotential expansion of the fermionic checkerboard (upper) and the honeycomb (lower) models for $L_x=L_y=6$. The normalized overlap $V^j$ is plotted against $\lambda$, where $H_{\mathrm{int}}=\lambda H_{\mathrm{NN}}+(1-\lambda )H_{\mathrm{NNN}}$, so that we have the NNN limit on the left and the NN limit on the right. For the CB model, we see that the NN and NNN terms exhibit marked differences in their pseudopotential expansions, with the MT symmetry-conserving part of its NNN term consisting almost exclusively of the $V^1$ and $V^3$ terms. This can be understood by studying its distribution of matrix elements (Fig. 5). For comparison, the PP coefficients $V^1$ to $V^9$ of the QH Coulomb interaction are plotted as dashed lines. The NNN interaction has a larger $V^1$ coefficient and smaller $V^3,V^5,...$ coefficients than the Coulomb interaction, and is hence even more likely to exhibit Laughlin ground state.

Figure 5

Plots of normalized $|H_p|$ for the NN (left) and NNN (center) terms of the fermionic CB model. The horizontal and vertical axes represent $l_1-l_2$ and $l_1+l_2$, respectively. Regions with relatively large $|H_P|$ are colored red. $U^1$ (right) is plotted for comparison. The NNN term evidently bears more resemblance to $U^1$.

Figure 6

The pseudopotential expansion of the fermionic (solid line) and bosonic (dashed line) HC models for $L_x=L_y=6$. The PP coefficient $V^j$ is plotted against $\lambda$, where $H_{\mathrm{int}}=\lambda H_{\mathrm{NN}}+(1-\lambda )H_{\mathrm{NNN}}$, so that we have the NNN limit on the left and the NN limit on the right. The bosonic PP coefficients are in general closer to each other, with $V^0$ not larger than $V^2$ for some values of $\lambda$.

Figure 7

The two types of MT breaking hoppings. Top: A hopping process that changes the c.m. position. Bottom: Two hopping processes that preserve the c.m. position but break magnetic translation symmetry because the hoppings at different c.m. position $n$ (solid and hollow arrows) have different amplitude.

Figure 8

Plot of ${\delta }_n^2$ for $L_x=L_y=6$ for different model Hamiltonians. As evident from the dominance of the peak at $n=0$, the amount of MT symmetry breaking largely stems from c.m. conserving terms. There is little difference between the degree of MT symmetry breaking in the different models.

Figure 9

Graphs of $b^{2j+1}$ from Eq. (C4) for $j=0,4,10,17$ with $\kappa =\frac{1}{6}$. As $j$ increases, the main region of contribution of $b^{2j+1}$ shifts in the direction of larger $\left|l\right|$.

Figure 10

The eigenvalues of the overlap matrix of $U^1,U^3,U^5,U^7$, and $U^9$ as a function of the system size $L=L_x=L_y$. Indeed, we observe orthogonality when $L\ge 6$, in excellent agreement with the bound $2\sqrt{9}=6$ from Eq. (D3).